Discussions

At the bottom of the bottle, there must be a tiny triangle with a number and/or some letters.
If you tell us what's the number (and/or the letters), we will be able to know exactly what type of plastic it is.

The "2" is the important number - that means the plastic is HDPE (high density polyethylene).

It is a thermoplastic (so it can be melted without changing it into a new substance). It is resistant to many solvents, so solvent-based glues (like those used for joining PVC pipes) will not work well. You can weld it with heat, as long as you are careful not to get it too hot (molten HDPE is flammable).

I would try "spot welding" with a soldering iron, or laying sheets on a heat-resistant surface, overlapping the edges and gently running a soldering iron down the overlap to heat-seal them.

If you have access to both side of the joint, make like the factory and hot-press. I'm tooling around with an istructable on it now, but the gist of the process is to overlap, press with some sort of metal die against a smooth metal surface, heat the die. If you don't have access to both sides, it is much tougher, and the plastic on the far side of the joint may want to curl away from you.

say you want to connect 2 together
form it with low heat (hot air or water) to the shape you want before joining
put the pieces together
fill with sand to make support from inside (so it won't warp)
heat with clothes iron thru aluminium foil to make it even or use hot glue
it won't warp too much if its filled and you press it to shape from outside immediately after applying hot glue and putting together

I'd be weary of any heat-based methods. I once tried to hot glue parts of a milk bottle, and it warped pretty badly. Mind you, the pieces were cut out of the side; so if you were to weld 2 bottoms together, you might have better luck.